math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

Alternatives: 15

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \cosh im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (sin re) (cosh im)))

C

double code(double re, double im) {
	return sin(re) * cosh(im);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sin(re) * cosh(im)
end function

Java

public static double code(double re, double im) {
	return Math.sin(re) * Math.cosh(im);
}

Python

def code(re, im):
	return math.sin(re) * math.cosh(im)

Julia

function code(re, im)
	return Float64(sin(re) * cosh(im))
end

MATLAB

function tmp = code(re, im)
	tmp = sin(re) * cosh(im);
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[Cosh[im], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

   8. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

   10. sub0-neg N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

   11. cosh-undef N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

   12. associate-*r* N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

   13. metadata-eval N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   21. lower-cosh.f64 100.0

       \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \sin re \]

   3. *-lft-identity N/A

      \[\leadsto \color{blue}{\cosh im} \cdot \sin re \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   5. lower-*.f64 100.0

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]
7. Add Preprocessing

Alternative 2: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (*
       (-
        (*
         (* (- (* -0.006944444444444444 (* im im)) 0.08333333333333333) im)
         im)
        0.16666666666666666)
       (* re re))
      re
      re)
     (if (<= t_0 1.0)
       (*
        (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0)
        (sin re))
       (*
        (* 0.5 re)
        (fma
         im
         im
         (fma
          (pow im 4.0)
          (fma 0.002777777777777778 (* im im) 0.08333333333333333)
          2.0)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((((-0.006944444444444444 * (im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * (re * re)), re, re);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = (0.5 * re) * fma(im, im, fma(pow(im, 4.0), fma(0.002777777777777778, (im * im), 0.08333333333333333), 2.0));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(-0.006944444444444444 * Float64(im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * Float64(re * re)), re, re);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, fma((im ^ 4.0), fma(0.002777777777777778, Float64(im * im), 0.08333333333333333), 2.0)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(-0.006944444444444444 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + N[(N[Power[im, 4.0], $MachinePrecision] * N[(0.002777777777777778 * N[(im * im), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 72.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{144} \cdot {im}^{2} - \frac{1}{12}\right) - \frac{1}{6}\right), re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      21. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      9. lower-*.f64 99.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 2.7%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot 2 \]
   7. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   8. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

   9. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left(2 + {im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left({im}^{2} \cdot \left(1 + {im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{\left({im}^{2} \cdot 1 + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right)} + 2\right) \]

       3. *-rgt-identity N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\left(\color{blue}{{im}^{2}} + {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right)\right) + 2\right) \]

       4. associate-+l+ N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\left({im}^{2} + \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \left(\color{blue}{im \cdot im} + \left({im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, {im}^{2} \cdot \left({im}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)\right) + 2\right)} \]

       7. associate-*r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \color{blue}{\left({im}^{2} \cdot {im}^{2}\right) \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right)} + 2\right) \]

       8. pow-sqr N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \color{blue}{{im}^{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 2\right) \]

       9. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, {im}^{\color{blue}{4}} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {im}^{2}\right) + 2\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \color{blue}{\mathsf{fma}\left({im}^{4}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2\right)}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left(\color{blue}{{im}^{4}}, \frac{1}{12} + \frac{1}{360} \cdot {im}^{2}, 2\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \color{blue}{\frac{1}{360} \cdot {im}^{2} + \frac{1}{12}}, 2\right)\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \color{blue}{\mathsf{fma}\left(\frac{1}{360}, {im}^{2}, \frac{1}{12}\right)}, 2\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \left(\frac{1}{2} \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(\frac{1}{360}, \color{blue}{im \cdot im}, \frac{1}{12}\right), 2\right)\right) \]

       15. lower-*.f64 70.8

           \[\leadsto \left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, \color{blue}{im \cdot im}, 0.08333333333333333\right), 2\right)\right) \]

   11. Applied rewrites 70.8%

       \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), 2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, \mathsf{fma}\left({im}^{4}, \mathsf{fma}\left(0.002777777777777778, im \cdot im, 0.08333333333333333\right), 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (*
       (-
        (*
         (* (- (* -0.006944444444444444 (* im im)) 0.08333333333333333) im)
         im)
        0.16666666666666666)
       (* re re))
      re
      re)
     (if (<= t_0 1.0)
       (*
        (fma (fma 0.041666666666666664 (* im im) 0.5) (* im im) 1.0)
        (sin re))
       (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((((-0.006944444444444444 * (im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * (re * re)), re, re);
	} else if (t_0 <= 1.0) {
		tmp = fma(fma(0.041666666666666664, (im * im), 0.5), (im * im), 1.0) * sin(re);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(-0.006944444444444444 * Float64(im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * Float64(re * re)), re, re);
	elseif (t_0 <= 1.0)
		tmp = Float64(fma(fma(0.041666666666666664, Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(-0.006944444444444444 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(N[(N[(0.041666666666666664 * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 72.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{144} \cdot {im}^{2} - \frac{1}{12}\right) - \frac{1}{6}\right), re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      21. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}, {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      9. lower-*.f64 99.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 73.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 62.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
   (if (<= t_1 (- INFINITY))
     (fma
      (*
       (-
        (*
         (* (- (* -0.006944444444444444 (* im im)) 0.08333333333333333) im)
         im)
        0.16666666666666666)
       (* re re))
      re
      re)
     (if (<= t_1 1.0)
       (* t_0 (fma im im 2.0))
       (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(((((((-0.006944444444444444 * (im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * (re * re)), re, re);
	} else if (t_1 <= 1.0) {
		tmp = t_0 * fma(im, im, 2.0);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(-0.006944444444444444 * Float64(im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * Float64(re * re)), re, re);
	elseif (t_1 <= 1.0)
		tmp = Float64(t_0 * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(-0.006944444444444444 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(t$95$0 * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 72.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{144} \cdot {im}^{2} - \frac{1}{12}\right) - \frac{1}{6}\right), re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 98.9

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 98.9%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 73.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 62.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (*
       (-
        (*
         (* (- (* -0.006944444444444444 (* im im)) 0.08333333333333333) im)
         im)
        0.16666666666666666)
       (* re re))
      re
      re)
     (if (<= t_0 1.0)
       (* 1.0 (sin re))
       (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp(-im) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(((((((-0.006944444444444444 * (im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * (re * re)), re, re);
	} else if (t_0 <= 1.0) {
		tmp = 1.0 * sin(re);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(-0.006944444444444444 * Float64(im * im)) - 0.08333333333333333) * im) * im) - 0.16666666666666666) * Float64(re * re)), re, re);
	elseif (t_0 <= 1.0)
		tmp = Float64(1.0 * sin(re));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(N[(-0.006944444444444444 * N[(im * im), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(1.0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 72.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in re around inf

      \[\leadsto \mathsf{fma}\left({re}^{2} \cdot \left({im}^{2} \cdot \left(\frac{-1}{144} \cdot {im}^{2} - \frac{1}{12}\right) - \frac{1}{6}\right), re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      21. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{1} \cdot \sin re \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

      \[\leadsto \color{blue}{1} \cdot \sin re \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 73.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 62.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.006944444444444444 \cdot \left(im \cdot im\right) - 0.08333333333333333\right) \cdot im\right) \cdot im - 0.16666666666666666\right) \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{elif}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;1 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;t\_0 \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0
         (fma
          (fma
           (fma 0.001388888888888889 (* im im) 0.041666666666666664)
           (* im im)
           0.5)
          (* im im)
          1.0)))
   (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 1.0)
     (* t_0 (sin re))
     (*
      (fma
       (pow re 3.0)
       (- (* 0.008333333333333333 (* re re)) 0.16666666666666666)
       re)
      t_0))))

C

double code(double re, double im) {
	double t_0 = fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0);
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 1.0) {
		tmp = t_0 * sin(re);
	} else {
		tmp = fma(pow(re, 3.0), ((0.008333333333333333 * (re * re)) - 0.16666666666666666), re) * t_0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 1.0)
		tmp = Float64(t_0 * sin(re));
	else
		tmp = Float64(fma((re ^ 3.0), Float64(Float64(0.008333333333333333 * Float64(re * re)) - 0.16666666666666666), re) * t_0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$0 * N[Sin[re], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[re, 3.0], $MachinePrecision] * N[(N[(0.008333333333333333 * N[(re * re), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + re), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 1

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      21. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   5. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

      14. lower-*.f64 92.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   7. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

   if 1 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

      8. lift-exp.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

      11. cosh-undef N/A

          \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

      14. cosh-0 N/A

          \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

      15. *-commutative N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      16. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

      17. cosh-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      18. exp-0 N/A

          \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

      19. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

      20. exp-0 N/A

          \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

      21. lower-cosh.f64 100.0

          \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \sin re \]

      3. *-lft-identity N/A

         \[\leadsto \color{blue}{\cosh im} \cdot \sin re \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

      5. lower-*.f64 100.0

         \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   7. Taylor expanded in im around 0

      \[\leadsto \sin re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \]

      4. +-commutative N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      11. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

      13. unpow2 N/A

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

      14. lower-*.f64 87.8

          \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \]

   9. Applied rewrites 87.8%

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \]

   10. Taylor expanded in re around 0

       \[\leadsto \color{blue}{\left(re \cdot \left(1 + {re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right)\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(re \cdot \color{blue}{\left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + 1\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{\left(re \cdot \left({re}^{2} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)\right) + re \cdot 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(re \cdot {re}^{2}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right)} + re \cdot 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\left(re \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{re}^{3}} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + re \cdot 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({re}^{3} \cdot \left(\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}\right) + \color{blue}{re}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{re}^{3}}, \frac{1}{120} \cdot {re}^{2} - \frac{1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2} - \frac{1}{6}}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({re}^{3}, \color{blue}{\frac{1}{120} \cdot {re}^{2}} - \frac{1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({re}^{3}, \frac{1}{120} \cdot \color{blue}{\left(re \cdot re\right)} - \frac{1}{6}, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]

       12. lower-*.f64 72.0

           \[\leadsto \mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \color{blue}{\left(re \cdot re\right)} - 0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \]

   12. Applied rewrites 72.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({re}^{3}, 0.008333333333333333 \cdot \left(re \cdot re\right) - 0.16666666666666666, re\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.24:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.24)
   (* (* (fma -0.08333333333333333 (* re re) 0.5) re) (fma im im 2.0))
   (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.24) {
		tmp = (fma(-0.08333333333333333, (re * re), 0.5) * re) * fma(im, im, 2.0);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.24)
		tmp = Float64(Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re) * fma(im, im, 2.0));
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.24], N[(N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.23999999999999999

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 80.1

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 80.1%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{12}, {re}^{2}, \frac{1}{2}\right)} \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{12}, \color{blue}{re \cdot re}, \frac{1}{2}\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

      6. lower-*.f64 55.5

         \[\leadsto \left(\mathsf{fma}\left(-0.08333333333333333, \color{blue}{re \cdot re}, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   if 0.23999999999999999 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 79.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 48.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.24:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im, re, re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.02)
   (fma (* -0.16666666666666666 (* re re)) re re)
   (fma (* (* (fma (* im im) 0.041666666666666664 0.5) im) im) re re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((-0.16666666666666666 * (re * re)), re, re);
	} else {
		tmp = fma(((fma((im * im), 0.041666666666666664, 0.5) * im) * im), re, re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re);
	else
		tmp = fma(Float64(Float64(fma(Float64(im * im), 0.041666666666666664, 0.5) * im) * im), re, re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * re + re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 83.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot {re}^{2}, re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 12.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 87.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in re around 0

      \[\leadsto \mathsf{fma}\left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right), re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.5%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im, re, re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(im \cdot im, 0.041666666666666664, 0.5\right) \cdot im\right) \cdot im, re, re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.02)
   (fma (* -0.16666666666666666 (* re re)) re re)
   (* (fma (* (fma (* 0.041666666666666664 im) im 0.5) im) im 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((-0.16666666666666666 * (re * re)), re, re);
	} else {
		tmp = fma((fma((0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re);
	else
		tmp = Float64(fma(Float64(fma(Float64(0.041666666666666664 * im), im, 0.5) * im), im, 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(N[(N[(N[(N[(0.041666666666666664 * im), $MachinePrecision] * im + 0.5), $MachinePrecision] * im), $MachinePrecision] * im + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 83.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot {re}^{2}, re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 12.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 87.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right), im \cdot im, 1\right) \cdot re \]
   11. Step-by-step derivation

   12. Applied rewrites 66.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot im, im, 0.5\right) \cdot im, im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.24:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right) \cdot re\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) 0.24)
   (fma (* -0.16666666666666666 (* re re)) re re)
   (* (fma (* (* im im) 0.041666666666666664) (* im im) 1.0) re)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= 0.24) {
		tmp = fma((-0.16666666666666666 * (re * re)), re, re);
	} else {
		tmp = fma(((im * im) * 0.041666666666666664), (im * im), 1.0) * re;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= 0.24)
		tmp = fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re);
	else
		tmp = Float64(fma(Float64(Float64(im * im) * 0.041666666666666664), Float64(im * im), 1.0) * re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.24], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(N[(N[(N[(im * im), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.23999999999999999

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 89.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot {re}^{2}, re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 45.1%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \]

   if 0.23999999999999999 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 79.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \color{blue}{re} \]

   9. Taylor expanded in im around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {im}^{2}, im \cdot im, 1\right) \cdot re \]
   10. Step-by-step derivation

   11. Applied rewrites 48.4%

       \[\leadsto \mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right) \cdot re \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq 0.24:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(im \cdot im\right) \cdot 0.041666666666666664, im \cdot im, 1\right) \cdot re\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))) -0.02)
   (fma (* -0.16666666666666666 (* re re)) re re)
   (* (* 0.5 re) (fma im im 2.0))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp(-im) + exp(im))) <= -0.02) {
		tmp = fma((-0.16666666666666666 * (re * re)), re, re);
	} else {
		tmp = (0.5 * re) * fma(im, im, 2.0);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.02)
		tmp = fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re);
	else
		tmp = Float64(Float64(0.5 * re) * fma(im, im, 2.0));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(im * im + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < -0.0200000000000000004

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

      13. lower-sin.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

      16. lower-sin.f64 83.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

   5. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

   9. Taylor expanded in im around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot {re}^{2}, re, re\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 12.7%

       \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \]

   if -0.0200000000000000004 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{\left({im}^{2} + 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \left(\color{blue}{im \cdot im} + 2\right) \]

      3. lower-fma.f64 79.4

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   5. Applied rewrites 79.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\mathsf{fma}\left(im, im, 2\right)} \]

   6. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 59.4

         \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]

   8. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot \mathsf{fma}\left(im, im, 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \mathsf{fma}\left(im, im, 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.001388888888888889 (* im im) 0.041666666666666664)
    (* im im)
    0.5)
   (* im im)
   1.0)
  (sin re)))

C

double code(double re, double im) {
	return fma(fma(fma(0.001388888888888889, (im * im), 0.041666666666666664), (im * im), 0.5), (im * im), 1.0) * sin(re);
}

Julia

function code(re, im)
	return Float64(fma(fma(fma(0.001388888888888889, Float64(im * im), 0.041666666666666664), Float64(im * im), 0.5), Float64(im * im), 1.0) * sin(re))
end

Wolfram

code[re_, im_] := N[(N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

   8. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

   10. sub0-neg N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

   11. cosh-undef N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

   12. associate-*r* N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

   13. metadata-eval N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   21. lower-cosh.f64 100.0

       \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

5. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \cdot \sin re \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \cdot \sin re \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \cdot \sin re \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \cdot \sin re \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \cdot \sin re \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \cdot \sin re \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \cdot \sin re \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \cdot \sin re \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

   14. lower-*.f64 91.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \cdot \sin re \]

7. Applied rewrites 91.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \cdot \sin re \]

8. Final simplification 91.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \cdot \sin re \]
9. Add Preprocessing

Alternative 13: 91.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (*
  (sin re)
  (fma (fma (* 0.001388888888888889 (* im im)) (* im im) 0.5) (* im im) 1.0)))

C

double code(double re, double im) {
	return sin(re) * fma(fma((0.001388888888888889 * (im * im)), (im * im), 0.5), (im * im), 1.0);
}

Julia

function code(re, im)
	return Float64(sin(re) * fma(fma(Float64(0.001388888888888889 * Float64(im * im)), Float64(im * im), 0.5), Float64(im * im), 1.0))
end

Wolfram

code[re_, im_] := N[(N[Sin[re], $MachinePrecision] * N[(N[(N[(0.001388888888888889 * N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision] + 0.5), $MachinePrecision] * N[(im * im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\sin re \cdot \left(\frac{1}{2} \cdot \left(e^{0 - im} + e^{im}\right)\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{0 - im} + e^{im}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(e^{im} + e^{0 - im}\right)}\right) \]

   7. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{e^{im}} + e^{0 - im}\right)\right) \]

   8. lift-exp.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + \color{blue}{e^{0 - im}}\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{0 - im}}\right)\right) \]

   10. sub0-neg N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right)\right) \]

   11. cosh-undef N/A

       \[\leadsto \sin re \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot \cosh im\right)}\right) \]

   12. associate-*r* N/A

       \[\leadsto \sin re \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot 2\right) \cdot \cosh im\right)} \]

   13. metadata-eval N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{1} \cdot \cosh im\right) \]

   14. cosh-0 N/A

       \[\leadsto \sin re \cdot \left(\color{blue}{\cosh 0} \cdot \cosh im\right) \]

   15. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   16. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\cosh 0 \cdot \cosh im\right) \cdot \sin re} \]

   17. cosh-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   18. exp-0 N/A

       \[\leadsto \left(\color{blue}{e^{0}} \cdot \cosh im\right) \cdot \sin re \]

   19. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(e^{0} \cdot \cosh im\right)} \cdot \sin re \]

   20. exp-0 N/A

       \[\leadsto \left(\color{blue}{1} \cdot \cosh im\right) \cdot \sin re \]

   21. lower-cosh.f64 100.0

       \[\leadsto \left(1 \cdot \color{blue}{\cosh im}\right) \cdot \sin re \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right) \cdot \sin re} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 \cdot \cosh im\right)} \cdot \sin re \]

   3. *-lft-identity N/A

      \[\leadsto \color{blue}{\cosh im} \cdot \sin re \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

   5. lower-*.f64 100.0

      \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\sin re \cdot \cosh im} \]

7. Taylor expanded in im around 0

   \[\leadsto \sin re \cdot \color{blue}{\left(1 + {im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sin re \cdot \color{blue}{\left({im}^{2} \cdot \left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin re \cdot \left(\color{blue}{\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right)\right) \cdot {im}^{2}} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2} + {im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right), {im}^{2}, 1\right)} \]

   4. +-commutative N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{{im}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) + \frac{1}{2}}, {im}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}\right) \cdot {im}^{2}} + \frac{1}{2}, {im}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {im}^{2}, {im}^{2}, \frac{1}{2}\right)}, {im}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {im}^{2} + \frac{1}{24}}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {im}^{2}, \frac{1}{24}\right)}, {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{im \cdot im}, \frac{1}{24}\right), {im}^{2}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   11. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), \color{blue}{im \cdot im}, \frac{1}{2}\right), {im}^{2}, 1\right) \]

   13. unpow2 N/A

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, im \cdot im, \frac{1}{24}\right), im \cdot im, \frac{1}{2}\right), \color{blue}{im \cdot im}, 1\right) \]

   14. lower-*.f64 91.4

       \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), \color{blue}{im \cdot im}, 1\right) \]

9. Applied rewrites 91.4%

   \[\leadsto \sin re \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, im \cdot im, 0.041666666666666664\right), im \cdot im, 0.5\right), im \cdot im, 1\right)} \]

10. Taylor expanded in im around inf

    \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {im}^{2}, im \cdot im, \frac{1}{2}\right), im \cdot im, 1\right) \]
11. Step-by-step derivation

12. Applied rewrites 91.2%

    \[\leadsto \sin re \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(im \cdot im\right), im \cdot im, 0.5\right), im \cdot im, 1\right) \]
13. Add Preprocessing

Alternative 14: 33.5% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (fma (* -0.16666666666666666 (* re re)) re re))

C

double code(double re, double im) {
	return fma((-0.16666666666666666 * (re * re)), re, re);
}

Julia

function code(re, im)
	return fma(Float64(-0.16666666666666666 * Float64(re * re)), re, re)
end

Wolfram

code[re_, im_] := N[(N[(-0.16666666666666666 * N[(re * re), $MachinePrecision]), $MachinePrecision] * re + re), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \color{blue}{\sin re + {im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{im}^{2} \cdot \left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) + \sin re} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re\right) \cdot {im}^{2}} + \sin re \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right) + \frac{1}{2} \cdot \sin re, {im}^{2}, \sin re\right)} \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot \sin re + \frac{1}{24} \cdot \left({im}^{2} \cdot \sin re\right)}, {im}^{2}, \sin re\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot \sin re + \color{blue}{\left(\frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin re \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)}, {im}^{2}, \sin re\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) \cdot \sin re}, {im}^{2}, \sin re\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {im}^{2} + \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {im}^{2}, \frac{1}{2}\right)} \cdot \sin re, {im}^{2}, \sin re\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{im \cdot im}, \frac{1}{2}\right) \cdot \sin re, {im}^{2}, \sin re\right) \]

   13. lower-sin.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \color{blue}{\sin re}, {im}^{2}, \sin re\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, im \cdot im, \frac{1}{2}\right) \cdot \sin re, \color{blue}{im \cdot im}, \sin re\right) \]

   16. lower-sin.f64 85.7

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \color{blue}{\sin re}\right) \]

5. Applied rewrites 85.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot \sin re, im \cdot im, \sin re\right)} \]

6. Taylor expanded in re around 0

   \[\leadsto re \cdot \color{blue}{\left(1 + \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right) + {re}^{2} \cdot \left(\frac{-1}{6} \cdot \left({im}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {im}^{2}\right)\right) - \frac{1}{6}\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 26.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, im \cdot im, -0.08333333333333333\right) \cdot \left(im \cdot im\right) - 0.16666666666666666, re \cdot re, \left(\mathsf{fma}\left(0.041666666666666664, im \cdot im, 0.5\right) \cdot im\right) \cdot im\right), \color{blue}{re}, re\right) \]

9. Taylor expanded in im around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot {re}^{2}, re, re\right) \]
10. Step-by-step derivation

11. Applied rewrites 33.6%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(re \cdot re\right), re, re\right) \]
12. Add Preprocessing

Alternative 15: 26.3% accurate, 28.8× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot 2 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) 2.0))

C

double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * 2.0d0
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * 2.0;
}

Python

def code(re, im):
	return (0.5 * re) * 2.0

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * 2.0;
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

3. Taylor expanded in im around 0

   \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{2} \]
4. Step-by-step derivation

5. Applied rewrites 50.2%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{2} \]

6. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot re\right)} \cdot 2 \]
7. Step-by-step derivation

   1. lower-*.f64 26.0

      \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]

8. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot re\right)} \cdot 2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))